L(s) = 1 | + i·3-s + (2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (1 + 2i)15-s − 2i·17-s + 2·21-s − 4i·23-s + (3 − 4i)25-s − i·27-s − 8·31-s − 2i·33-s + (−2 − 4i)35-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s + 0.436·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s − 1.43·31-s − 0.348i·33-s + (−0.338 − 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33864 - 0.827325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33864 - 0.827325i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00753669308419175057550088676, −9.193411808798071220306016277263, −8.254969611591929950200229108632, −7.48323700282678448355579266359, −6.27774383454896964527410242383, −5.37964832598748357281358339416, −4.77142167373993736883932077938, −3.50680404998244847127583535008, −2.43558900516617993360117622150, −0.72102067387008897618786772945,
1.76623382470639652667722352824, 2.41364340619497992275745377862, 3.77205334529209662907691509218, 5.24833189380140533966738785239, 5.88131568020044699660890229835, 6.78755993818593172158689370167, 7.46062972812106738853262156018, 8.715158220990760030641822257168, 9.212099851975808375776571483064, 10.12096110965081070063231957866